Algebraic topology

Algebraic topology is a branch of mathematics that studies topological spaces using the tools of abstract algebra. The primary goal is to identify algebraic invariants, which are properties that remain unchanged under certain transformations. These invariants help mathematicians classify spaces up to homeomorphism or homotopy equivalence. While the field mainly uses algebra to solve topological problems, the reverse is also true. For instance, topology provides a convenient proof that any subgroup of a free group is itself a free group. This illustrates how the two fields support one another through a deep, structural connection.

To understand the mechanism of this field, one must look at how spaces are converted into algebraic structures. Mathematicians seek a correspondence between spaces and groups that respects the relationship of homeomorphism. This allows complex statements about shapes to be recast as statements about groups. Groups possess a highly manageable structure, which makes these proofs much easier to complete. Two major methods for this conversion are homotopy theory and the study of homology and cohomology groups. Homotopy theory uses fundamental groups to provide basic information about a space's structure. However, these groups are often nonabelian and can be quite difficult to manipulate.

In contrast, homology and cohomology groups offer a different approach. Homology is a general procedure that associates a sequence of abelian groups or modules with a mathematical object. Because these groups are abelian, they are often easier to work with, especially when they are finitely generated. Cohomology is a more refined method that assigns algebraic invariants to a space using a cochain complex. It arises from the algebraic dualization of the homology construction. While homology looks at the basic structure, cohomology provides a more detailed algebraic description through the study of cochains, cocycles, and coboundaries.

There are several distinct types of objects and spaces studied within this discipline. Manifolds are topological spaces that resemble Euclidean space near every point. Common examples include the plane, the sphere, and the torus. Some manifolds, like the Klein bottle or the real projective plane, cannot be embedded in three dimensions and require four dimensions instead. Another area is knot theory, which studies mathematical knots. Unlike a common shoelace knot, a mathematical knot is an embedding of a circle in three-dimensional Euclidean space where the ends are joined. Two knots are considered equivalent only if one can be transformed into the other through ambient isotopy, which involves deforming the string without cutting it.

Mathematicians also construct spaces using complexes. A simplicial complex is built by gluing together points, line segments, triangles, and higher-dimensional counterparts. A related concept is the CW complex, which was introduced by J. H. C. Whitehead. CW complexes are broader than simplicial complexes and possess better categorical properties for homotopy theory. These structures allow for computational work while maintaining a combinatorial nature. Historically, the field was known as combinatorial topology because it focused on how spaces were constructed from simpler parts. By the 1920s and 1930s, the emphasis shifted toward investigating spaces through algebraic correspondences, leading to the modern name.

The history of the field is marked by significant individual contributions. Georges de Rham was a pioneer in cohomology research. He demonstrated that different approaches, such as de Rham cohomology and simplicial homology, were interrelated. Specifically, he showed that for a closed, oriented manifold, the Betti numbers derived from these different methods were identical. In the 1950s, Samuel Eilenberg and Norman Steenrod expanded this work by generalizing these approaches. They defined homology and cohomology as functors equipped with natural transformations. They also proved that an axiomatization based on these rules uniquely characterized the theory.

The significance of algebraic topology is seen in its many profound applications. It can be used to prove the fundamental theorem of algebra by using the fundamental group of a circle. It also provides the basis for the Brouwer fixed point theorem, which states that every continuous map from a unit n-disk to itself has a fixed point. In geometry, the "hairy ball theorem" describes how an n-sphere admits a nowhere-vanishing continuous unit vector field only if n is odd. Furthermore, the field connects to broader concepts like category theory. In fact, the notions of category, functor, and natural transformation originated within the constructions of algebraic topology.